A Dual Method for Computing Power Transfer Distribution Factors.
Ronellenfitsch, H., Timme, M., Witthaut, D.
IEEE Transactions on Power Systems 32 (2), 1007-1015 (2017), https://doi.org/10.1109/TPWRS.2016.2589464.Abstract: Power Transfer Distribution Factors (PTDFs) play a crucial role in power grid security analysis, planning, and redispatch.
Fast calculation of the PTDFs is therefore of great importance. In this paper, we present a non-approximative dual method of computing PTDFs.
It uses power flows along topological cycles of the network but still relies on simple matrix algebra.
At the core, our method changes the size of the matrix that needs to be inverted to calculate the PTDFs from N×N, where N is the number of buses,
to (L-N+1)×(L-N+1), where L is the number of lines and L-N+1 is the number of independent cycles (closed loops) in the network while remaining mathematically fully equivalent.
For power grids containing a relatively small number of cycles, the method can offer a speedup of numerical calculations.

A universal order parameter for synchrony in networks of limit cycle oscillators.
Schröder, M., Timme, M., Witthaut, D.
preprint at: arXiv:1704.04130, Manuscript under review for Chaos (2017)Abstract: We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks.
First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence
over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial
phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators.
We rigorously proof that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network.
Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where
different types of synchrony are possible, including biological networks and power grids.

Can Distribution Grids Significantly Contribute to Transmission Grids’ Voltage Management?
Auer, S. Steinke, F. Chunsen, W. Szabo, A. Sollacher, R.
IEEE ISGT (2017), DOI: 10.1109/ISGTEurope.2016.7856194.Abstract: Power generation in Germany is currently transitioning from a system based on large, central, thermal power
plants to one that heavily relies on small, decentral, mostly renewable power generators. This development poses the question
how transmission grids’ reactive power demand for voltage management, covered by central power plants today, can be supplied in the future.
In this work, we estimate the future technical potential of such an approach for the whole of Germany. For a 100% renewable
electricity scenario we set the possible reactive power supply in comparison with the reactive power requirements that are needed
to realize the simulated future transmission grid power flows.
Since an exact calculation of distribution grids’ reactive power potential is difficult due to the unavailability of detailed grid
models on such scale, we optimistically estimate the potential by assuming a scaled, averaged distribution grid model connected
to each of the transmission grid nodes.
We find that for all except a few transmission grid nodes, the required reactive power can be fully supplied from the modelled
distribution grids. This implies that – even if our estimate is overly optimistic – distributed reactive power provisioning will
be a technical solution for many future reactive power challenges.

Curing critical links in oscillator networks as power flow models.
Rohden, M., Witthaut, D., Timme, M., Meyer-Ortmanns, H.
New J. Phys. 19, 013002 (2017).Abstract: Modern societies crucially depend on the robust supply with electric energy so that blackouts of power grids can have far reaching consequences.
Typically, large scale blackouts take place after a cascade of failures: the failure of a single infrastructure component,
such as a critical transmission line, results in several subsequent failures that spread across large parts of the network.
Improving the robustness of a network to prevent such secondary failures is thus key for assuring a reliable power supply.
In this article we analyze the nonlocal rerouting of power flows after transmission line failures for a simplified AC power grid model and compare different strategies to improve network robustness.
We identify critical links in the grid and compute alternative pathways to quantify the grid's redundant capacity and to find bottlenecks along the pathways.
Different strategies are developed and tested to increase transmission capacities to restore stability with respect to transmission line failures.
We show that local and nonlocal strategies typically perform alike: one can equally well cure critical links by providing backup capacities locally or
by extending the capacities of bottleneck links at remote locations.

Cycle flows and multistabilty in oscillatory networks: an overview.
Manik, D., Timme, M., Witthaut, D.
submitted to Chaos, preprint at https://arxiv.org/abs/1611.09825 (2017).Abstract: The functions of many networked systems in physics, biology or engineering rely on a coordinated or synchronized dynamics of its constituents.
In power grids for example, all generators must synchronize and run at the same frequency and their phases need to appoximately lock to guarantee a steady power flow.
Here, we analyze the existence and multitude of such phase-locked states. Focusing on edge and cycle flows instead of the nodal phases we derive rigorous results on the existence and number of such states.
Generally, multiple phase-locked states coexist in networks with strong edges, long elementary cycles and a homogeneous distribution of natural frequencies or power injections,
respectively. We offer an algorithm to systematically compute multiple phase- locked states and demonstrate some surprising dynamical consequences of multistability.

Deciphering the imprint of topology on nonlinear dynamical network stability.
Nitzbon, J., Schultz, P., Heitzig, J., Kurths, J., Hellmann, F.
New Journal of Physics, 19 033029 (2017).Abstract: Coupled oscillator networks show complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes
with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system,
its survivability, as well as its asymptotic behaviour, its basin stability. By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary,
desynchronized oscillators which are rotating with a frequency different from their natural one. They occur almost exclusively after perturbations at nodes with specific topological properties.
More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability.
We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties,
thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks.
From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a
better performance in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrised to be representative for power grids,
we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields.

Dual theory of transmission line outages.
Ronellenfitsch, G., Manik, D., Hörsch, J., Brown, T., Witthaut, D.
IEEE Transactions on Power Systems, Volume: PP, Issue: 99 (2017).Abstract: A new graph dual formalism is presented for the analysis of line outages in electricity networks.
The dual formalism is based on a consideration of the flows around closed cycles in the network.
After some exposition of the theory is presented, a new formula for the computation of Line Outage Distribution Factors (LODFs) is derived,
which is not only computationally faster than existing methods, but also generalizes easily for multiple line outages and arbitrary changes to line series reactance.
In addition, the dual formalism provides new physical insight for how the effects of line outages propagate through the network.
For example, in a planar network a single line outage can be shown to induce monotonically decreasing flow changes, which are mathematically equivalent to an electrostatic dipole field.

Escape Routes, Weak Links, and Desynchronization in Fluctuation-driven Networks.
Schaefer, B., Matthiae, M., Zhang, X., Rohden, M., Timme, M., Witthaut D.
accepted at Phys. Rev. E (Rapid Communications) (2017).Abstract: Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system.
Here, we demonstrate how increased fluctuations, reduced damping and reduced intertia may undermine the dynamical robustness of power grid networks. ^
Focusing on fundamental noise models, we derive analytic insights into which factors limit the dynamic robustness and how fluctuations may induce a system escape from an operating state.
Moreover, we identify weak links in the grid that make it particularly vulnerable to fluctuations.
These results thereby not only contribute to a theoretical understanding of how fluctuations act on distributed network dynamics, they may also help designing future renewable energy systems to be more robust.

Extreme prices in electricity balancing markets from an approach of statistical physics.
Mureddu, M., Meyer-Ortmanns, H.
submitted (2017).Abstract: An increase in energy production from renewable energy sources is viewed as a crucial achievement in most industrialized countries.
The higher variability of power production via renewables leads to a rise in ancillary service costs over the power system, in particular costs within the electricity balancing markets,
mainly due to an increased number of extreme price spikes.
This study focuses on forecasting the behavior of price and volumes of the Italian balancing market in the presence of an increased share of renewable energy sources.
Starting from configurations of load and power production, which guarantee a stable performance, we implement fluctuations in the load and in renewables;
in particular we artificially increase the contribution of renewables as compared to conventional power sources to cover the total load.
We then forecast the amount of provided energy in the balancing market and its fluctuations, which are induced by production and consumption.
Within an approach of agent based modeling we estimate the resulting energy prices and costs.
While their average values turn out to be only slightly affected by an increased contribution from renewables, the probability for extreme price events is shown to increase along with undesired peaks in the costs.

From State Estimation to Network Reconstruction.
Basiri, F., Casadiego, J., Timme, M., Witthaut, M.
submitted to Phys. Rev. Applied., preprint at https://arxiv.org/abs/1701.09084 (2017).Abstract: We develop methods to efficiently reconstruct the topology and line parameters of a power grid from the measurement of nodal variables.
We propose two compressed sensing algorithms that minimize the amount of necessary measurement resources by exploiting network sparsity, symmetry of connections and potential prior knowledge about the connectivity.
The algorithms are reciprocal to established state estimation methods, where nodal variables are estimated from few measurements given the network structure.
Hence, they enable an advanced grid monitoring where both state and structure of a grid are subject to uncertainties or missing information.

Linear Optimal Power Flow Using Cycle Flows.
Hörsch, J., Ronellenfitsch, H., Witthaut, D., Brown, T.
arXiv:1704.01881 (2017).Abstract: Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC)
load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches.
Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous.
In this article we provide a review of existing methods and describe new formulations, which express the loading constraints directly in terms of the flows themselves,
using a decomposition of the graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations,
showing that one of the new formulations of the LOPF solves up to 20 times faster than the angle formulation using commercial linear programming solvers, with an average speed-up
of factor 3 for the standard networks considered here. The speed-up is largest for networks with many nodes and decentral generators throughout the network, which is highly relevant
given the rise of distributed renewable generation.

Long-term impacts of a coal phase-out in Germany as part of a greenhouse gas mitigation strategy.
Heinrichs, h., Markewitz, P.
Applied Energy, Volume 192, 15 April 2017, Pages 234-246, ISSN 0306-2619, https://doi.org/10.1016/j.apenergy.2017.01.065.Abstract: Germany appears set to miss its CO2 reduction target in 2020.
As a result, ideas for additional political measures have been put forward. One such idea involves an early phase-out of coal-fired power plants.
However, the possible impacts of such a phase-out on the energy system have not yet been fully analyzed.
We therefore apply a German energy system model to analyze these impacts. To do so, we calculate three different scenarios.
The first represents a business-as-usual scenario, while the second takes a coal phase-out into account.
The third scenario has to achieve the same CO2 reduction as the second without being forced to implement a coal phase-out.
Our three scenarios show that a definitive coal phase-out by 2040 would result in only a relatively small amount of additional CO2.
However, an equal CO2 reduction can be obtained using a different strategy and slightly lower costs.
In the latter scenario, the additional costs are also distributed more evenly across the sectors.
The sensitivities analyzed show the robustness of the conclusions drawn.
In summary, this analysis outlines what consequences could arise by excluding several options in parallel from a technology portfolio.

More Homogeneous Wind Conditions Under Strong Climate Change Decrease the Potential for Inter-State Balancing of Electricity in Europe.
Wohland, J., Reyers, M., Weber, J., Witthaut, D.
preprint at: https://doi.org/10.5194/esd-2017-48, 2017. Manuscript under review for Earth System Dynamics.Abstract: Limiting anthropogenic climate change requires the fast decarbonisation of the electricity system.
Renewable electricity generation is determined by the weather and is hence subject to climate change.
We simulate the operation of a coarse-scale fully-renewable European electricity system based on downscaled high resolution climate data from EURO-CORDEX.
Following a high emission pathway (RCP8.5), we find a robust increase of backup needs in Europe until the end of the 21st century.
The absolute increase of the backup needs is almost independent of potential grid expansion, leading to the paradoxical effect that relative impacts
of climate change increase in a highly interconnected European system.
The increase is rooted in more homogeneous wind conditions over Europe resulting in extensive parallel generation shortfalls.
Our results are strengthened by comparison with a large CMIP5 ensemble using an approach based on Circulation Weather Types.

Network susceptibilities: theory and applications.
Manik, D., Rohden, M., Ronellenfitsch, H., Zhang, X., Hallerberg, S., Witthaut, D., Timme, M.
Phys. Rev. E 95, 012310 (2017).Abstract: We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes.
We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively.
We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models.
Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena.
The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.

Optimal heterogeneity in a simplified highly renewable European electricity system.
Eriksen, E., Schwenk-Nebbe, L., Tranberg, B., Brown, T., Greiner, M.
Energy, https://doi.org/10.1016/j.energy.2017.05.170 (2017).Abstract: The resource quality and the temporal generation pattern of variable renewable energy sources vary significantly across Europe.
In this paper spatial distributions of renewable assets are explored which exploit this heterogeneity to lower the total system costs for a high level of renewable electricity in Europe.
Several intuitive heuristic algorithms, optimal portfolio theory and a local search algorithm are used to find optimal distributions of renewable generation capacities that minimise the total costs of backup,
transmission and renewable capacity simultaneously. Using current cost projections, an optimal heterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea,
which results in average electricity costs that are up to 11% lower than for a homogeneous reference distribution of renewables proportional to each country's mean load.
The reduction becomes even larger, namely 18%, once the transmission capacities are put to zero in the homogeneous reference distribution. Heuristic algorithms to distribute renewable capacity based
on each country's wind and solar capacity factors are shown to provide a satisfactory approximation to fully optimised renewable distributions, while maintaining the benefits of transparency and comprehensibility.
The sensitivities of the results to changing costs of solar generation and gas supply as well as to the possible cross-sectoral usage of unavoidable curtailment energy are also examined.

Potentials and limits to basin stability estimation.
Schultz, P., Menck P.J., Heitzig, J., Kurths, J.
New Journal of Physics, Volume 19 (2017).Abstract: Stability assessment methods for dynamical systems have recently been complemented by basin stability and derived measures,
i.e. probabilistic statements whether systems remain in a basin of attraction given a distribution of perturbations.
Their application requires numerical estimation via Monte Carlo sampling and integration of differential equations.
Here, we analyse the applicability of basin stability to systems with basin geometries that are challenging for this numerical method,
having fractal basin boundaries and riddled or intermingled basins of attraction.
We find that numerical basin stability estimation is still meaningful for fractal boundaries but reaches its limits for riddled basins with holes.

Power-functional network.
Sun, Y., Kurths, J., Zhan, M.
Chaos 27, 083116 (2017); doi: http://dx.doi.org/10.1063/1.4995361 (2017).Abstract: Power grids and their properties have been studied broadly in many aspects.
In this paper, we propose a novel concept, power-flow-based power grid, as a typical power-functional network, based on the calculation of power flow distribution from power electrical engineering.
We compare it with structural networks based on the shortest path length and effective networks based on the effective electrical distance and study the relationship among these three kinds of networks.
We find that they have roughly positive correlations with each other, indicating that in general any close nodes in the topological structure are actually connected in function.
However, we do observe some counter-examples that two close nodes in a structural network can have a long distance in a power-functional network,
namely, two physically connected nodes can actually be separated in function.
In addition, we find that power grids in the structural network tend to be heterogeneous, whereas those in the effective and power-functional networks tend to be homogeneous.
These findings are expected to be significant not only for power grids but also for various other complex networks.
Unveiling the core structure-function relationship has become a key objective in many disciplines.
Borrowing the key concept in brain connectome, functional brain network, here we propose a similar concept, power-functional network (FN), to describe the functional connection in power systems,
and especially, the so-called power-flow-based power grid, to describe the functional connection for transporting active power in power systems.
The calculation of the power-flow-based power grid is mainly based on the power flow tracing, which has been extensively used in power electrical engineering and power economic market.
Its difference with the structural network from the shortest path length in the graph theory and the effective network (EN) for the effective electrical distance is studied here.
We expect that this study is helpful for our understanding of the structure-function relationship not only in power grids but also in various other flow complex systems.

Stability of meshed DC microgrids using probabilistic analysis.
Strenge, L., Kirchhoff, H., Ndow, L. G., Hellmann, F.
DC Microgrids (ICDCM), IEEE Second International Conference on DC Microgrids (2017).Abstract: We analyze the stability of meshed DC microgrids using the proposed Probabilistic Analysis of Deterministic Systems (PADeS).
By sampling a random distribution of networks as well as disturbances, we make probabilistic statements valid for two- (ring) and three-valent (meshed) networks.
The producing nodes are droop controlled sources and the consuming nodes are modeled as constant power loads (CPLs). ^
We introduce basin stability and survivability as probabilistic stability measures within the PADeS framework and illustrate that it has a large potential for understanding the properties of DC microgrids and beyond.
To show this, we run numerical experiments with 128-node networks and 1000 samples for the system's probability (a) to return to a stable equilibrium point after a perturbation (basin stability)
and (b) to remain within defined operational bounds for disturbances such as sudden load power increase, line outages and short circuits at random nodes or power lines (survivability).

Stability of Synchrony against Local Intermittent Fluctuations in Tree-like Power Grids.
Auer, S., Hellmann, F., Krause, M., Kurths, J.
Accepted Chaos, ArXiv:1702.08707 (2017).Abstract: 80% of all Renewable Energy Power in Germany is installed in tree-like distribution grids.
Intermittent power fluctuations from such sources introduce new dynamics into the lower grid layers.
At the same time, distributed resources will have to contribute to stabilize the grid against these fluctuations in the future.
In this paper, we model a system of distributed resources as oscillators on a tree-like, lossy power grid and its ability to withstand desynchronization from localized intermittent renewable infeed.
We find a remarkable interplay of network structure and the position of the node at which the fluctuations are fed in.
An important precondition for our findings is the presence of losses in distribution grids.
Then, the most network central node splits the network into branches with different influence on network stability.
Troublemakers, i.e. nodes at which fluctuations are especially exciting the grid, tend to be downstream branches with high net power outflow.
For low coupling strength, we also find branches of nodes vulnerable to fluctuations anywhere in the network.
These network regions can be predicted at high confidence using an eigenvector based network measure taking the turbulent nature of perturbations into account.
While we focus here on tree-like networks, the observed effects also appear, albeit less pronounced, for weakly meshed grids.
On the other hand the observed effects disappear for lossless power grids often studied in the complex systems literature.

The Benefits of Cooperation in a Highly Renewable European Electricity Network.
Schlachtberger, D., Brown, T., Schramm, S., Greiner, M.
arXiv:1704.05492, accepted by Energy (2017).Abstract: To reach ambitious European CO2 emission reduction targets, most scenarios of future European electricity systems rely on large shares of wind and solar photovoltaic power generation.
We interpolate between two concepts for balancing the variability of these renewable sources: balancing at continental scales using the transmission grid and balancing locally with storage.
This interpolation is done by systematically restricting transmission capacities from the optimum level to zero.
We run techno-economic cost optimizations for the capacity investment and dispatch of wind, solar, hydroelectricity, natural gas power generation and transmission, as well as storage options such as pumped-hydro,
battery, and hydrogen storage. The simulations assume a 95% CO2 emission reduction compared to 1990, and are run over a full historical year of weather and electricity demand for 30 European countries.
In the cost-optimal system with high levels of transmission expansion, energy generation is dominated by wind (65%) and hydro (15%), with average system costs comparable to today's system.
Restricting transmission shifts the balance in favour of solar and storage, driving up costs by a third. As the restriction is relaxed, 85% of the cost benefits of the
optimal grid expansion can be captured already with only 44% of the transmission volume.

The Contribution of Different Electric Vehicle Control Strategies to Dynamical Grid Stability.
Auer, S., Roos, C., Heitzig, J., Hellmann, F., Kurths, J.
arXiv:1708.03531 (2017).Abstract: A major challenge for power grids with a high share of renewable energy systems (RES), such as island grids, is to provide frequency stability in the face of renewable fluctuations.
In this work we evaluate the ability of electric vehicles (EV) to provide distributed primary control and to eliminate frequency peaks.
To do so we for the first time explicitly model the network structure and incorporate non-Gaussian, strongly intermittent fluctuations typical for RES.
We show that EVs can completely eliminate frequency peaks. Using threshold randomization we further demonstrate that demand synchronization effects and battery stresses can be greatly reduced.
In contrast, explicit frequency averaging has a strong destabilizing effect, suggesting that the role of delays in distributed control schemes requires further studies.
Overall we find that distributed control outperforms central one. The results are robust against a further increase in renewable power production and fluctuations.

The role of spatial scale in joint optimisations of generation and transmission for European highly renewable scenarios.
Hörsch, J., Brown, T.
accepted to 14th International Conference on the European Energy Market - EEM 2017 (2017).Abstract: The effects of the spatial scale on the results of the optimisation of transmission and generation capacity in Europe
are quantified under a 95% CO2 reduction compared to 1990 levels, interpolating between one-node-per-country solutions and many-nodes-per-country.
The trade-offs that come with higher spatial detail between better exposure of transmission bottlenecks, exploitation of sites with good renewable resources
(particularly wind power) and computational limitations are discussed.
It is shown that solutions with no grid expansion beyond today's capacities are only around 20% more expensive than with cost-optimal grid expansion.

- 2016 -

Cascading failures in ac electricity grids.
Rohden, M. Jung, D. Tamrakar, S. Kettemann, S.
Phys. Rev. E 94, 032209 – Published 9 September 2016.Abstract: Sudden failure of a single transmission element in a power grid can induce a domino effect of cascading failures,
which can lead to the isolation of a large number of consumers or even to the failure of the entire grid. Here we present results
of the simulation of cascading failures in power grids, using an alternating current (AC) model. We first apply this model to a regular square grid topology.
For a random placement of consumers and generators on the grid, the probability to find more than a certain number of unsupplied consumers decays as a power
law and obeys a scaling law with respect to system size. Varying the transmitted power threshold above which a transmission line fails does not seem
to change the power law exponent q˜1.6. Furthermore, we study the influence of the placement of generators and consumers on the number of affected consumers
and demonstrate that large clusters of generators and consumers are especially vulnerable to cascading failures. As a real-world topology we consider the German
high-voltage transmission grid. Applying the dynamic AC model and considering a random placement of consumers, we find that the probability to disconnect more
than a certain number of consumers depends strongly on the threshold. For large thresholds the decay is clearly exponential,
while for small ones the decay is slow, indicating a power law decay.

Critical links and nonlocal rerouting in complex supply networks.
Witthaut, D., Rohden, M., Zhang, X., Hallerberg, S., Timme, M.
Phys. Rev. Lett. 116, 138701 (2016).Abstract: Link failures repeatedly induce large-scale outages in power grids and other supply networks.
Yet, it is still not well understood which links are particularly prone to inducing such outages.
Here we analyze how the nature and location of each link impact the network’s capability to maintain a stable supply.
We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure.
They are determined via a link’s redundant capacity and a renormalized linear response theory we derive.
These criteria outperform the critical link prediction based on local measures such as loads.
The results not only further our understanding of the physics of supply networks in general.
As both criteria are available before any outage from the state of normal operation,
they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design.

Delocalization of Phase Disturbances and the Stability of AC Electricity Grids.
Kettemann, S.
Phys. Rev. E 94, 062311 (2016), published December 2016.Abstract: In order to study how local disturbances affect the ac grid stability, we start from nonlinear power balance equations and map them to complex linear wave equations.
Having obtained stationary solutions with phases i at generator and consumer nodes i, we next study the dynamics of deviations.
Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid.
We find the parametric dependence of diffusion constant D. We apply the same solution strategy to general grid topologies and analyze their stability against local perturbations.
The perturbation remains either localized or becomes delocalized, depending on grid topology, power capacity, and distribution of consumers and generator power Pi.
Delocalization is found to increase the lifetime of perturbations and thereby their influence on grid stability, whereas localization results in an exponentially fast decay of perturbations at all grid sites.
These results may therefore lead to new strategies to control the stability of electricity grids.

Deciphering the imprint of topology on nonlinear dynamical network stability.
Nitzbon, J., Schultz, P., Heitzig, J., Kurths, J., Hellmann, F.
arXiv:1612.03654 (2016).Abstract: Coupled oscillator networks show a complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes
with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system, its survivability,
as well as its asymptotic behaviour, its basin stability.
By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary, desynchronized oscillators which are rotating with a frequency different from their natural one.
They occur almost exclusively after perturbations at nodes with specific topological properties.
More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability.
We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties,
thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks.
From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a better performance
in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrized to be representative for power grids,
we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields.

Flexibility Mechanisms and Pathways to a Highly Renewable US Electricity Future.
Frew, B.A., Becker, S., Dvorak, M.J., Andresen, G.B., Jacobson, M.Z.
Energy 101 (2016) 65-78.Abstract: This study explores various scenarios and flexibility mechanisms to integrate high penetrations of renewable energy into the US (United States) power grid.
A linear programming model – POWER (Power system Optimization With diverse Energy Resources) – is constructed and used to (1) quantify flexibility cost-benefits of geographic aggregation,
renewable overgeneration, storage, and flexible electric vehicle charging, and (2) compare pathways to a fully renewable electricity system.
Geographic aggregation provides the largest flexibility benefit with ~5–50% cost savings, but each region's contribution to the aggregate RPS (renewable portfolio standard)
target is disproportionate, suggesting the need for regional-and-resource-specific RPS targets.
Electric vehicle charging yields a lower levelized system cost, revealing the benefits of demand-side flexibility.
However, existing demand response price structures may need adjustment to encourage optimal flexible load in highly renewable systems.
Two scenarios with RPS targets from 20% to 100% for the US (peak load ~729 GW) and California (peak load ~62 GW) find each RPS target feasible from a planning perspective,
but with 2× the cost and 3× the overgeneration at a 100% versus 80% RPS target. Emission reduction cost savings for the aggregated
US system with an 80% versus 20% RPS target are roughly $200 billion/year, outweighing the $80 billion/year cost for the same RPS range.

Generalised flow tracing for the analysis of networked renewable electricity systems.
Hörsch, J., Schäfer, M., Becker, S., Schramm, S., Greiner, M.
submitted for review to IEEE Transactions on Power Systems (2016).Abstract: Flow allocation methods represent a valuable tool set to analyze the power flows in networked electricity systems.
Based on this flow allocation, the costs associated with the usage of the underlying network infrastructure can be assigned to the users of the electricity system.
This paper presents a generalization of the flow tracing method that is applicable to arbitrary compositions of inflow appearing naturally in aggregated networks.
The composition of inflow is followed from net-generating sources through the network and assigns corresponding shares of the total power flow as well as of the outflow to the net-consuming sinks.
We showcase the analytical power of this method for a scenario based on the IEEE 118 bus network and emphasize the need of appropriate aggregating measures,
which allow to integrate over whole time series of fluctuating flow patterns.

Interaction Control to Synchronize Non-synchronizable Networks.
Schroeder, M., Chakraborty, S., Witthaut, D., Nagler, J., Timme, M.
Scientific Reports 6, 37142 (2016).Abstract: Synchronization constitutes one of the most fundamental collective dynamics across networked systems and often underlies their function.
Whether a system may synchronize depends on the internal unit dynamics as well as the topology and strength of their interactions.
For chaotic units with certain interaction topologies synchronization might be impossible across all interaction strengths, meaning that these networks are non-synchronizable.
Here we propose the concept of interaction control, generalizing transient uncoupling, to induce desired collective dynamics in complex networks
and apply it to synchronize even such non-synchronizable systems.
After highlighting that non-synchronizability prevails for a wide range of networks of arbitrary size, we explain how a simple binary control may localize interactions in state space and thereby synchronize networks.
Intriguingly, localizing interactions by a fixed control scheme enables stable synchronization across all connected networks regardless of topological constraints.
Interaction control may thus ease the design of desired collective dynamics even without knowledge of the networks’ exact interaction topology and consequently have
implications for biological and self-organizing technical systems.

Islanding the power grid on the transmission level: less connections for more security.
Mureddu, M. Caldarelli, G. Damiano, A. Scala, A. Meyer-Ortmanns, H.
Scientific Reports 6, Article number: 34797 (2016).Abstract: Islanding is known as a management procedure of the power system that is implemented at the distribution level
to preserve sensible loads from outages and to guarantee the continuity in electricity supply, when a high amount of distributed generation occurs.
In this paper we study islanding on the level of the transmission grid and shall show that it is a suitable measure to enhance energy security and grid resilience.
We consider the German and Italian transmission grids. We remove links either randomly to mimic random failure events, or according to a topological characteristic,
their so-called betweenness centrality, to mimic an intentional attack and test whether the resulting fragments are self-sustainable. We test this option via the tool of optimized DC power flow equations.
When transmission lines are removed according to their betweenness centrality, the resulting islands have a higher chance of being dynamically self-sustainable than for a random removal.
Less connections may even increase the grid’s stability. These facts should be taken into account in the design of future power grids.

Local vs. global redundancy -- trade-offs between resilience against cascading failures and frequency stability.
Plietzsch, A., Schultz, P., Heitzig, J., Kurths, J.
European Physical Journal-Special Topics, 225(3), 551-568, 2016, DOI: 10.1140/epjst/e2015-50137-4.Abstract: When designing or extending electricity grids, both frequency stability and resilience against cascading failures have to be considered
amongst other aspects of energy security and economics such as construction costs due to total line length.
Here, we compare an improved simulation model for cascading failures with state-of-the-art simulation models for short-term grid dynamics.
Random ensembles of realistic power grid topologies are generated using a recent model that allows for a tuning of global vs local redundancy.
The former can be measured by the algebraic connectivity of the network, whereas the latter can be measured by the networks transitivity.
We show that, while frequency stability of an electricity grid benefits from a global form of redundancy,
resilience against cascading failures rather requires a more local form of redundancy and further analyse the corresponding trade-off.

Long-range response in ac electricity grids.
Jung, D., Kettemann, S.
Journal: Phys. Rev. E 94, 012307 (2016), DOI: 10.1103/PhysRevE.94.012307.Abstract: Local changes in the topology of electricity grids can cause overloads far away from the disturbance
[D. Witthaut and M. Timme, Eur. Phys. J. B 86, 377 (2013)],
making the prediction of the robustness against changes in the topology—for example, caused by power outages or grid extensions—a challenging task. The impact of single-line
additions on the long-range response of dc electricity grids has recently been studied
[D. Labavic, R. Suciu, H. Meyer-Ortmanns, and S. Kettemann, Eur. Phys. J.: Spec. Top. 223, 2517 (2014)].
By solving the real part of the static ac load flow equations, we conduct a similar investigation for ac grids. In a regular two-dimensional
grid graph with cyclic boundary conditions, we find a power law decay for the change of power flow as a function of distance to the disturbance
over a wide range of distances. The power exponent increases and saturates for large system sizes. By applying the same analysis to the German transmission
grid topology, we show that also in real-world topologies a long-ranged response can be found.

Optimising the European transmission system for 77% renewable electricity by 2030.
Brown, T., Schierhorn, P.-P., Tröster, E., Ackermann, T.
IET-RPG, Volume 10, Issue 1, p. 3–9 (Jan 2016).Abstract: Local changes in the topology of electricity grids can cause overloads far away from the disturbance
[D. Witthaut and M. Timme, Eur. Phys. J. B 86, 377 (2013)], making the prediction of the robustness against changes in the topology—for example,
caused by power outages or grid extensions—a challenging task. The impact of single-line additions on the long-range response of dc electricity grids
has recently been studied [D. Labavic, R. Suciu, H. Meyer-Ortmanns, and S. Kettemann, Eur. Phys. J.: Spec. Top. 223, 2517 (2014)].
By solving the real part of the static ac load flow equations, we conduct a similar investigation for ac grids.
In a regular two-dimensional grid graph with cyclic boundary conditions, we find a power law decay for the change of power flow
as a function of distance to the disturbance over a wide range of distances. The power exponent increases and saturates for large system sizes.
By applying the same analysis to the German transmission grid topology, we show that also in real-world topologies a long-ranged response can be found.

Power flow tracing in complex networks.
Schäfer, M., Hempel, S., Hörsch, J., Tranberg, S., Schramm, S. and Greiner M.
In: Schramm, S. and Schäfer, M. (Eds.) New Horizons in Fundamental Physics. FIAS Interdisciplinary Science Series (2016).Abstract: The increasing share of decentralized renewable power generation represents a challenge to the current and future energy system.
Providing a geographical smoothing effect, long-range power transmission plays a key role for the system integration of these fluctuating resources.
However, the build-up and operation of the necessary network infrastructure incur costs which have to be allocated to the users of the system.
Flow tracing techniques, which attribute the power flow on a transmission line to the geographical location of its generation and consumption,
represent a valuable tool set to design fair usage and thus cost allocation schemes for transmission investments.
In this article, we introduce a general formulation of the flow tracing method and apply it to a simplified model of a highly renewable European electricity system.
We review a statistical usage measure which allows to integrate network usage information for longer time series, and illustrate this measure using an analytical test case.

Representation of the German transmission grid for Renewable Energy Sources.
Mureddu, M.
arXiv:1612.05532 (2016).Abstract: The increasing impact of fossil energy generation on the Earth ecological balance is pointing to the need of a transition in power generation technology
towards the more clean and sustainable Renewable Energy Sources (RES).
This transition is leading to new paradigms and technologies useful for the effective energy transmission and distribution, which take into account the RES stochastic power output.
In this scenario, the availability of up to date and reliable datasets regarding topological and operative parameters of power systems in presence of RES are needed, for both proposing and testing new solutions.
In this spirit, I present here a dataset regarding the German 380 KV grid which contains fully DC Power Flow operative states of the grid in the presence of various amounts of RES share,
ranging from realistic up to 60\%, which can be used as reference dataset for both steady state and dynamical analysis.

Survivability of Deterministic Dynamical Systems.
Hellmann, F., Schultz, P., Grabow, C., Heitzig, J., Kurths, J.
Scientific Reports, 6, 29654 (2016), DOI: 10.1038/srep29654.Abstract: The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research.
It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability:
Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states.
We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids.
We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids.
Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.

Synchronizing noisy nonidentical oscillators by transient uncoupling.
Tandon, A., Schröder, M., Mannattil, M., Timme, M., Chakraborty, S.
Chaos 26:094817 (2016).Abstract: Synchronization is the process of achieving identical dynamics among coupled identical units.
If the units are different from each other, their dynamics cannot become identical; yet, after transients,
there may emerge a functional relationship between them—a phenomenon termed “generalized synchronization.”
Here, we show that the concept of transient uncoupling, recently introduced for synchronizing identical units,
also supports generalized synchronization among nonidentical chaotic units. Generalized synchronization can be achieved
by transient uncoupling even when it is impossible by regular coupling. We furthermore demonstrate that
transient uncoupling stabilizes synchronization in the presence of common noise.
Transient uncoupling works best if the units stay uncoupled whenever the driven orbit
visits regions that are locally diverging in its phase space.
Thus, to select a favorable uncoupling region, we propose an intuitive method that measures the local divergence
at the phase points of the driven unit's trajectory by linearizing the flow and subsequently suppresses the divergence by uncoupling.

Taming Instabilities in Power Grid Networks by Decentralized Control.
Schaefer, B., Grabow, C., Auer, S., Kurths, J., Witthaut, D., Timme, M.
Eur. Phys. J. ST 225, 569 (2016).Abstract: Renewables will soon dominate energy production in our electric power system.
And yet, how to integrate renewable energy into the grid and the market is still a subject of major debate.
Decentral Smart Grid Control (DSGC) was recently proposed as a robust and decentralized approach to balance supply
and demand and to guarantee a grid operation that is both economically and dynamically feasible.
Here, we analyze the impact of network topology by assessing the stability of essential network motifs using both linear
stability analysis and basin volume for delay systems. Our results indicate that if frequency measurements are averaged
over sufficiently large time intervals, DSGC enhances the stability of extended power grid systems.
We further investigate whether DSGC supports centralized and/or decentralized power production and find it to be applicable to both.
However, our results on cycle-like systems suggest that DSGC favors systems with decentralized production.
Here, lower line capacities and lower averaging times are required compared to those with centralized production.

The impact of model detail on power grid resilience measures.
Auer, S., Kleis, K., Schultz, P., Kurths, J., Hellmann, F.
European Physical Journal-Special Topics, 225(3), 609-625, 2016, DOI: 10.1140/epjst/e2015-50265-9.Abstract: Extreme events are a challenge to natural as well as man-made systems.
For critical infrastructure like power grids, we need to understand their resilience against large disturbances.
Recently, new measures of the resilience of dynamical systems have been developed in the complex system literature.
Basin stability and survivability respectively assess the asymptotic and transient behavior of a system when subjected to arbitrary,
localized but large perturbations in frequency and phase. To employ these methods that assess power grid resilience, we need to choose
a certain model detail of the power grid. For the grid topology we considered the Scandinavian grid and an ensemble of power grids
generated with a random growth model. So far the most popular model that has been studied is the classical swing equation model
for the frequency response of generators and motors. In this paper we study a more sophisticated model of synchronous machines that also
takes voltage dynamics into account, and compare it to the previously studied model.
This model has been found to give an accurate picture of the long term evolution of synchronous machines in the engineering
literature for post fault studies. We find evidence that some stable fix points of the swing equation become unstable when we add voltage dynamics.
If this occurs the asymptotic behavior of the system can be dramatically altered, and basin stability estimates obtained with the swing
equation can be dramatically wrong. We also find that the survivability does not change significantly when taking the voltage dynamics into account.
Further, the limit cycle type asymptotic behaviour is strongly correlated with transient voltages that violate typical operational voltage bounds.
Thus, transient voltage bounds are dominated by transient frequency bounds and play no large role for realistic parameters.

The relevance of grid expansion under zonal markets.
Bertsch, J., Brown, T., Hagspiel, S., Just, L.
The Energy Journal, IAEE, Vol. 38, Number 5(2016).Abstract: The European electricity market design is based on zonal markets with uniform prices.
Hence, no differentiated locational price signals are provided within these zones.
If intra-zonal congestion occurs due to missing grid expansion, this market design reveals its inherent incompleteness,
and might lead to severe short and long-term distortions. In this paper, we study these distortions with a focus on the impact of restricted grid expansion under zonal markets.
Therefore, we use a long-term model of the European electricity system and restrict the allowed expansion of the transmission grid per decade.
We find that the combination of an incomplete market design and restricted grid expansion leads to a misallocation of generation capacities and the inability to transport electricity to where it is needed.
This results in an energy imbalance in some regions of up to 2-3% and difficulty when reaching envisaged political targets in the power sector.

Tweaking synchronization by connectivity modifications.
Schultz, P., Peron, T., Eroglu, D., Stemler, T., Ramirez Avila, G. M., Rodrigues, F. A., Kurths, J.
Physical Review E, 93(6), 062211, 2016, DOI: 10.1103/PhysRevE.93.062211.Abstract: Natural and man-made networks often possess locally treelike substructures. Taking such tree networks as our starting point,
we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition.
The first method adds single links that create cycles of a well-defined length. Following a topological approach, we introduce cycles of varying
length and analyze how this feature, as well as the position in the network, alters the synchronous behavior.
We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum,
dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes.
We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either discrete or continuous.
Here our main result is that a certain number of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization.

Backup flexibility classes in emerging large-scale renewable electricity systems.
Schlachtberger, D., Becker, S., Schramm, S., Greiner, M.
Energy Conversion and Management 4, 20 (2016).Abstract: High shares of intermittent renewable power generation in a European electricity system will require flexible backup power generation on the dominant diurnal,
synoptic, and seasonal weather timescales. The same three timescales are already covered by today’s dispatchable electricity generation facilities,
which are able to follow the typical load variations on the intra-day, intra-week, and seasonal timescales.
This work aims to quantify the changing demand for those three backup flexibility classes in emerging large-scale electricity systems,
as they transform from low to high shares of variable renewable power generation. A weather-driven modelling is used, which aggregates
eight years of wind and solar power generation data as well as load data over Germany and Europe, and splits the backup system required
to cover the residual load into three flexibility classes distinguished by their respective maximum rates of change of power output.
This modelling shows that the slowly flexible backup system is dominant at low renewable shares, but its optimized capacity decreases
and drops close to zero once the average renewable power generation exceeds 50% of the mean load.
The medium flexible backup capacities increase for modest renewable shares, peak at around a 40% renewable share,
and then continuously decrease to almost zero once the average renewable power generation becomes larger than 100% of the mean load.
The dispatch capacity of the highly flexible backup system becomes dominant for renewable shares beyond 50%, and reach their maximum around a 70% renewable share.
For renewable shares above 70% the highly flexible backup capacity in Germany remains at its maximum, whereas it decreases again for Europe.
This indicates that for highly renewable large-scale electricity systems the total required backup capacity can only be reduced if countries share their excess generation and backup power.

Cost-optimal design of a simplified, highly renewable pan-European electricity system.
Rodriguez, R.A., Becker, S., Greiner, M.
Energy 83, p. 658–668 (2015).Abstract: Based on a data-intensive weather-driven modelling approach, technically and economically optimal designs are derived for a simplified,
highly renewable pan-European electricity system, which minimise the need for backup energy, backup capacity, transmission capacity and the levelised
system cost of delivered electricity. The overall cost-optimal design, based on standard cost assumptions, relies on synchronised backup across the
transmission grid and comes with a renewable penetration of 50% with a rather high wind fraction of 94%. Given the current European electricity consumption,
this corresponds to 600 GW rated wind power capacities, 60 GW installed solar power capacities, 320 GW conventional backup power capacity, and about five times
today's installed transmission capacities. A sensitivity analysis reveals that the design and cost of the optimal system depend mostly on the assumed cost of
wind capacity and fuel for backup energy. Lower costs for wind capacity, higher costs for backup energy and usage of otherwise curtailed excess electricity
generation lead to a strong increase of the optimal renewable penetration. The sensitivity analysis is also used to find that a CO2 tax of over 100 €/ton
would be needed for the economic viability of carbon capture and sequestration.

Decentral Smart Grid Control.
Schäfer, B., Matthiae, M., Timme, M., Witthaut, D.
Journal: New Journal of Physics, 17, 015002 (2015), 10.1088/1367-2630/17/1/015002.Abstract: Stable operation of complex flow and transportation networks requires balanced supply and demand.
For the operation of electric power grids—due to their increasing fraction of renewable energy sources
—a pressing challenge is to fit the fluctuations in decentralized supply to the distributed and temporally varying demands.
To achieve this goal, common smart grid concepts suggest to collect consumer demand data, centrally evaluate them given
current supply and send price information back to customers for them to decide about usage.
Besides restrictions regarding cyber security, privacy protection and large required investments,
it remains unclear how such central smart grid options guarantee overall stability.
Here we propose a Decentral Smart Grid Control, where the price is directly linked to the local grid frequency at each customer.
The grid frequency provides all necessary information about the current power balance such that it is sufficient to match supply
and demand without the need for a centralized IT infrastructure.Weanalyze the performance and the dynamical stability of the power grid with such a control system.
Our results suggest that the proposed Decentral Smart Grid Control is feasible independent of effective measurement delays,
if frequencies are averaged over sufficiently large time intervals.

Delocalization of Phase Perturbations and the Stability of Electricity Grids.
Kettemann, S.
arXiv:1504.05525v2, subm. to Phys. Rev. Lett. (2015).Abstract: The energy transition towards an increased supply of renewable energy raises concerns that existing electricity grids,
built to connect few centralized large power plants with consumers, may become more difficult to control and stabilized with a rising number of
decentralized small scale generators. Here, we aim to study therefore, how local phase perturbations which may be caused by local power fluctuations,
affect the AC grid stability. To this end, we start from nonlinear power balance equations and map them to complex linear wave equations,
yielding stationary solutions with phases fi at generator and consumer sites i. Next, we study deviations from these stationary solutions.
Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid.
We derive the parametric dependence of diffusion constant D. We apply the same solution strategy to general grid topologies and analyse their
stability against local perturbations. The perturbation remains either localized or becomes delocalized, depending on grid topology,
power capacity and distribution of consumers and generators Pi. Delocalization is found to increase the lifetime of perturbations and
thereby their influence on grid stability, while localization results in an exponentiallyfast decay of perturbations at all grid sites.
These results may therefore lead to new strategies to control the stability of electricity grids.

Distribution planning and pricing in view of increasing shares of intermittent, renewable energy in Germany and Japan.
Brandstätt, C., Brunekreeft, G., Furusawa, K., Hattori, T.
Renewable Energy in Germany and Japan (2015).Abstract: In response to the global climate challenge many countries are faced with increasing shares of energy from renewable sources in their power supply. The integration of RES
(renewable energy sources) generation however entails technical as well as institutional challenges for power grids. This study relies on recent experiences of German distribution network
operators in network planning and network pricing and looks at their transferability to Japan. Distributed generation may cause problems of voltage variation and asset overloading in
conventional power grids. Technical solutions for these problems are available and well-known yet require considerable investments. The study presents regulatory incentives for network operators to
take efficient means to maintain supply quality. With distributed generation self-supplying customers may contribute too little to network cost and new generators and flexible consumers may cause
significant investment by uncoordinated siting and operation. An adequate pricing scheme can serve to sustainably finance the infrastructure while at the same time giving incentives to coordinate
network users. This study points out options for network charging in grids with high shares of distributed generation from renewable sources.

Focus on Networks, Energy and the Economy.
Timme, M., Kocarev, L., Witthaut, D.
New J. Phys. 17, 110201 (2015).Abstract: A sustainable and reliable energy supply constitutes a fundamental prerequisite for the future of our society.
The change to renewable sources comes with several systemic changes and includes, among others, smaller and more distributed producers
as well as stronger and less predictable fluctuations. Parallel developments such as the transition from conventional producers and
consumers to prosumers and the increasing number of electric vehicles add further complications. These changes require to extend and
upgrade currently existing power grids. Yet precisely how to achieve an effective, robustly operating (electric) energy system is far
from being understood. This focus issue aims to contribute to a number of these upcoming challenges from the perspective of self-organization
and the collective nonlinear dynamics of power grids, interacting economic factors as well as technical restrictions and opportunities for distributed systems.

Kuramoto dynamics in Hamiltonian systems.
Witthaut, D., Timme, M..
Journal: Physical Review E, 90, 032917 (2014). DOI: 10.1103/PhysRevE.90.032917.Abstract: The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators,
characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative)
system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds.
We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian action dynamics
of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics perturbed off the manifold
indicates the global synchronization transition point for finite N more precisely than the standard Kuramoto order parameter.
The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.

Localized vs. synchronized exports across a highly renewable pan-European transmission network.
Rodriguez, R.A., Dahl, M., Becker, S., Greiner, M.
Energy, Sustainability and Society 5:21 (2015).Abstract:
Background: A future, highly renewable electricity system will be largely based on fluctuating renewables.
The integration of wind and solar photovoltaics presents a major challenge. Transmission can be used to lower the need for complementary generation, which we term backup in this article.
Methods: Generation data based on historical weather data, combined with real load data, determine hourly mismatch timeseries for all European countries, connected by physical power flows.
Two localized export schemes determining the power flows are discussed, which export only renewable excess power, but no backup power,
and are compared to a synchronized export scheme, which exports renewable excess power and also backup power.
Results: Compared to no or very limited power transmission, unconstrained power flows across a highly renewable pan-European electricity network significantly reduce the overall amount of
required annual backup energy, but not necessarily the required backup capacities.
Conclusions: The reduction of the backup capacities turns out to be sensitive to the choice of export scheme. Results suggest that the synchronized export of local backup power
to other countries is important to significantly save on installed backup capacities.

Nonlocal effects and counter measures in cascading failures.
Witthaut, D. Timme, M.
Phys. Rev. E 92, 032809 (2015).Abstract: We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure.
It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads.
These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks.
Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges.
Nonlocal actions are generally required in networks that have a low redundancy and are thus especially vulnerable to cascades.

Power flow tracing in a simplified highly renewable European electricity network.
Tranberg, B., Thomsen, A., Rodriguez, R., Andresen, G., Schäfer, M., Greiner, M.
New Journal of Physics, 17(10):105002 (2015).Abstract: The increasing transmission capacity needs in a future energy system raise the question of how associated costs
should be allocated to the users of a strengthened power grid. In contrast to straightforward oversimplified methods,
a flow tracing based approach provides a fair and consistent nodal usage and thus cost assignment of transmission investments.
This technique follows the power flow through the network and assigns the link capacity usage to the respective sources or sinks
using a diffusion-like process, thus taking into account the underlying network structure and injection pattern.
As a showcase, we apply power flow tracing to a simplified model of the European electricity grid with a high share of renewable wind
and solar power generation, based on long-term weather and load data with an hourly temporal resolution.

Renewable electricity: Generation costs are not system costs.
Becker, S., Frew, B.A., Andresen, G.B., Jacobson, M.Z., Schramm, S., Greiner, M.
Energy 81, p. 437–445 (2015). preprint available at http://arxiv.org/abs/1412.4934Abstract: The transition to a future electricity system based primarily on wind and solar PV is examined for all regions in the contiguous US.
We present optimized pathways for the build-up of wind and solar power for least backup energy needs as well as for least cost obtained with a simplified,
lightweight model based on long-term high resolution weather-determined generation data. In the absence of storage, the pathway which achieves the best match
of generation and load, thus resulting in the least backup energy requirements, generally favors a combination of both technologies, with a wind/solar PV (photovoltaics)
energy mix of about 80/20 in a fully renewable scenario. The least cost development is seen to start with 100% of the technology with the lowest average generation costs first,
but with increasing renewable installations, economically unfavorable excess generation pushes it toward the minimal backup pathway.
Surplus generation and the entailed costs can be reduced significantly by combining wind and solar power, and/or absorbing excess generation,
for example with storage or transmission, or by coupling the electricity system to other energy sectors.

The Dynamics of Coalition Formation on Complex Networks.
Auer, S. Heitzig, J. Kornek, U. Schöll, E. Kurths, J.
Nature Scientific Reports 5, 13386; (2015). DOI: 10.1038/srep13386.Abstract: Complex networks describe the structure of many socio-economic systems. However, in studies of
decision-making processes the evolution of the underlying social relations are disregarded. In this
report, we aim to understand the formation of self-organizing domains of cooperation (“coalitions”)
on an acquaintance network. We include both the network’s influence on the formation of coalitions
and vice versa how the network adapts to the current coalition structure, thus forming a social
feedback loop. We increase complexity from simple opinion adaptation processes studied in earlier
research to more complex decision-making determined by costs and benefits, and from bilateral
to multilateral cooperation. We show how phase transitions emerge from such coevolutionary
dynamics, which can be interpreted as processes of great transformations. If the network adaptation
rate is high, the social dynamics prevent the formation of a grand coalition and therefore full
cooperation. We find some empirical support for our main results: Our model develops a bimodal
coalition size distribution over time similar to those found in social structures. Our detection and
distinguishing of phase transitions may be exemplary for other models of socio-economic systems
with low agent numbers and therefore strong finite-size effects.

The relevance of grid expansion under zonal markets.
Bertsch, J., Brown, T., Hagspiel, S., Just, L.
EWI Working Paper, No 15/07 (2015), accepted pending review in The Energy Journal.Abstract: The European electricity market design is based on zonal markets with uniform prices.
Locational price signals within these zones - necessary to ensure long-term efficiency - are not provided.
Speci cally, if intra-zonal congestion occurs due to missing grid expansion, the market design is
revealed as inherently incomplete. This might lead to severe, unwanted distortions of the electricity
market, both in the short- and in the long-term. In this paper, we study these distortions with
a speci c focus on the impact of restricted grid expansion under zonal markets. For this, we use
a long term fundamental dispatch and investment model of the European electricity system and
gradually restrict the allowed expansion of the transmission grid per decade. We nd that the combination
of an incomplete market design and restricted grid expansion leads to a misallocation of
generation capacities and the inability to transport electricity to where it is needed. Consequences
are severe and lead to load curtailment of up to 2-3 %. Moreover, missing grid expansion makes it
difficult and costly to reach envisaged energy targets in the power sector. Hence, we argue that in
the likely event of restricted grid expansion, either administrative measures or - presumably more
efficient - an adaptation of the current market design to include locational signals will become necessary.

Transient Uncoupling Induces Synchronization.
M. Schröder, M. Mannattil, D. Dutta, S. Chakraborty, M. Timme.
Phys. Rev. Lett. 115:054101 (2015).Abstract: Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines.
Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction
of their state space may synchronize even if fully coupled they do not. While for many standard systems coupling strengths need to be bounded to ensure synchrony,
transient uncoupling removes this bound and thus enables synchronization in an infinite range of effective coupling strengths.
The presented coupling scheme therefore opens up the possibility to induce synchrony in (biological or technical) systems whose parameters are fixed and cannot be modified continuously.

Value of Lost Load: An Efficient Economic Indicator for Power Supply Security? A Literature Review.
Schröder, T., Kuckshinrichs, W.
Front. Energy Res., 24 December 2015.Abstract: Security of electricity supply has become a fundamental requirement for well-functioning modern societies.
Because of its central position in all sections of society, the present paper considers the economic consequences of a power supply interruption.
The value of lost load (VoLL) is a monetary indicator expressing the costs associated with an interruption of electricity supply.
This paper reviews different methods for calculating VoLL, provides an overview of recently published studies, and presents suggestions to
increase the explanatory power and international comparability of VoLL.

Whose line is it anyway? Tracing the flows through Germany’s power grid.
Hörsch, J., Schäfer, M., Becker, S., Schramm, S., Greiner, M.
submitted for review and published as SDEWES proceedings (Sep 2015).

- 2014 -

A random growth model for power grids and other spatially embedded infrastructure networks.
Schultz, P., Heitzig, J., Kurths, J.
European Physical Journal-Special Topics, 223(12), 2593-2610, (2014), DOI: 10.1140/epjst/e2014-02279-6.Abstract: We propose a model to create synthetic networks that may also serve as a narrative of a certain kind of infrastructure network evolution.
It consists of an initialization phase with the network extending tree-like for minimum cost and a growth phase with an attachment rule giving a trade-off between cost-optimization and redundancy.
Furthermore, we implement the feature of some lines being split during the grid's evolution. We show that the resulting degree distribution has an exponential tail and may
show a maximum at degree two, suitable to observations of real-world power grid networks. In particular, the mean degree and the slope of the exponential decay can be controlled in partial independence.
To verify to which extent the degree distribution is described by our analytic form, we conduct statistical tests, showing that the hypothesis of an exponential tail is well-accepted for our model data.

Detours around basin stability in power networks.
Schultz, P., Heitzig, J., Kurths, J.
New Journal of Physics, 16, (2014), DOI: 10.1088/1367-2630/16/12/125001.Abstract: To analyse the relationship between stability against large perturbations and topological properties of a power transmission grid,
we employ a statistical analysis of a large ensemble of synthetic power grids, looking for significant statistical relationships between the single-node
basin stability measure and classical as well as tailormade weighted network characteristics. This method enables us to predict
poor values of single-node basin stability for a large extent of the nodes, offering a node-wise stability estimation at low computational cost.
Further, we analyse the particular function of certain network motifs to promote or degrade the stability of the system. Here we uncover the
impact of so-called detour motifs on the appearance of nodes with a poor stability score and discuss the implications for power grid design.

Impact of network topology on synchrony of oscillatory power grids.
Rohden, M., Sorge, A., Witthaut, D., Timme, M.
Chaos: an Interdisciplinary Journal of Nonlinear Science, 24(1), 013123, (2014).Abstract: Replacing conventional power sources by renewable sources in current power grids drastically alters their structure and functionality.
In particular, power generation in the resulting grid will be far more decentralized, with a distinctly different topology.
Here, we analyze the impact of grid topologies on spontaneous synchronization, considering regular, random, and small-world topologies and focusing
on the influence of decentralization. We model the consumers and sources of the power grid as second order oscillators.
First, we analyze the global dynamics of the simplest non-trivial (two-node) network that exhibit a synchronous (normal operation) state, a limit cycle (power outage),
and coexistence of both. Second, we estimate stability thresholds for the collective dynamics of small network motifs, in particular,
star-like networks and regular grid motifs. For larger networks, we numerically investigate decentralization scenarios finding that decentralization
itself may support power grids in exhibiting a stable state for lower transmission line capacities. Decentralization may thus be beneficial for power grids,
regardless of the details of their resulting topology. Regular grids show a specific sharper transition not found for random or small-world grids.

Supply networks: Instabilities without overload.
Manik, D., Witthaut, D., Schaefer, B., Matthiae, M., Sorge, A., Rohden, M., Katifori, E., Timme, M.
Journal: European Physical Journal Special Topics, 223, 2527 (2014). DOI: 10.1140/epjst/e2014-02274-y.Abstract: Supply and transport networks support much of our technical infrastructure as well as many biological processes.
Their reliable function is thus essential for all aspects of life. Transport processes involving quantities beyond the pure loads
exhibit alternative collective dynamical options compared to processes exclusively characterized by loads.
Here we analyze the stability and bifurcations in oscillator models describing electric power grids and
demonstrate that these networks exhibit instabilities without overloads.
This phenomenon may well emerge also in other sufficiently complex supply or transport networks, including biological transport processes.